Question

draw a graph of (f(x) = \frac{4x - 8}{-2x + 2}) by first placing the horizontal and vertical asymptotes, then plotting an additional point on the graph.

Explanation:

Step1: Find the vertical asymptote

Set the denominator equal to zero:

$$-2x + 2 = 0 \implies x = 1$$

Step2: Find the horizontal asymptote

Divide the leading coefficients of the numerator and denominator:

$$y = \frac{4}{-2} \implies y = -2$$

Step3: Find an additional point

Evaluate the function at \(x = 2\):

$$f(2) = \frac{4(2) - 8}{-2(2) + 2} = \frac{0}{-2} = 0$$

Answer:

The vertical asymptote is at \(x = 1\), the horizontal asymptote is at \(y = -2\), and an additional point on the graph is \((2, 0)\).